Average Error: 9.9 → 0.1
Time: 18.3s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{\frac{1}{x + 1}}{x} \cdot \frac{2}{x - 1}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{\frac{1}{x + 1}}{x} \cdot \frac{2}{x - 1}
double f(double x) {
        double r90670 = 1.0;
        double r90671 = x;
        double r90672 = r90671 + r90670;
        double r90673 = r90670 / r90672;
        double r90674 = 2.0;
        double r90675 = r90674 / r90671;
        double r90676 = r90673 - r90675;
        double r90677 = r90671 - r90670;
        double r90678 = r90670 / r90677;
        double r90679 = r90676 + r90678;
        return r90679;
}
↓
double f(double x) {
        double r90680 = 1.0;
        double r90681 = x;
        double r90682 = 1.0;
        double r90683 = r90681 + r90682;
        double r90684 = r90680 / r90683;
        double r90685 = r90684 / r90681;
        double r90686 = 2.0;
        double r90687 = r90681 - r90682;
        double r90688 = r90686 / r90687;
        double r90689 = r90685 * r90688;
        return r90689;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.9
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 9.9

    \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
  2. Using strategy rm

  3. Applied frac-sub26.1

    \[\leadsto \color{blue}{\frac{1 \cdot x - \left(x + 1\right) \cdot 2}{\left(x + 1\right) \cdot x}} + \frac{1}{x - 1}\]

  4. Applied frac-add25.8

    \[\leadsto \color{blue}{\frac{\left(1 \cdot x - \left(x + 1\right) \cdot 2\right) \cdot \left(x - 1\right) + \left(\left(x + 1\right) \cdot x\right) \cdot 1}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}}\]

  5. Simplified26.1

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 \cdot x - \left(x + 1\right) \cdot 2, x - 1, \left(\left(x + 1\right) \cdot x\right) \cdot 1\right)}}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\]

  6. Taylor expanded around 0 0.3

    \[\leadsto \frac{\color{blue}{2}}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\]
  7. Using strategy rm

  8. Applied *-un-lft-identity0.3

    \[\leadsto \frac{\color{blue}{1 \cdot 2}}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\]

  9. Applied times-frac0.1

    \[\leadsto \color{blue}{\frac{1}{\left(x + 1\right) \cdot x} \cdot \frac{2}{x - 1}}\]
  10. Using strategy rm

  11. Applied associate-/r*0.1

    \[\leadsto \color{blue}{\frac{\frac{1}{x + 1}}{x}} \cdot \frac{2}{x - 1}\]

  12. Final simplification0.1

    \[\leadsto \frac{\frac{1}{x + 1}}{x} \cdot \frac{2}{x - 1}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

  :herbie-target
  (/ 2 (* x (- (* x x) 1)))

  (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))))